Hi Ed,

In your three examples you have listed the same 10 digits but in different orders. It looks like you want the number of ways of listing the ten digits in different orders. These are permutations not combinations. The major difference is that when mathematicians use the term combinations, the order doesn't matter. Thus for example if you are looking for 3 digit combinations from the ten digits then 345, 831 and 098 are different combinations but 567, 675, and 756 are the same combination.

You want to know the number of ten digit permutations. Suppose you write one down, starting with the leftmost digit. You have 10 choices for the first digit you write. Whichever digit you use to start you can extend it to a 2 digit string in 9 different ways since you can't repeat the first digit. Thus there are $10 \times 9$ possible two digit permutations. Now each of these can be extended to a 3 digit permutation in 8 ways and hence there are $10 \times 9 \times 8$ possible three digit permutations.

Continue.

Penny